**5. Conclusions**

In this paper, the free vibration analysis of functionally graded shells is studied by using the ES-MITC3. Herein, the sti ffness matrices obtained based on the strain-smoothing technique over the smoothing domains associated with edges of MITC3 shell elements. The present approach uses a triangular element and hence much easily generated automatically, even for complicated geometries. The numerical results showed that the ES-MITC3 is a good agreemen<sup>t</sup> with the reference solutions, which are a requirement of high computational costs, such as ED555 [19] and a higher-order theory based on radial basis functions [17]. The ES-MITC3 presented herein is promising to be a simple and effective finite element method for analysis of functionally graded shells in practice.

The combination of finite element method (FEM) with an edge-based smoothed finite element method (ES-FEM) is very suitable for analyzing shell structures, especially for the complicated structures such as shell with reinforced sti ffeners, reinforced nano grapheme, micro shell structures, nano shell structures, and so on. This combination allows us to calculate exactly plate and shell structures with thin thicknesses (h = a/108) due to overcoming the shear locking phenomenon.

**Author Contributions:** Investigation, Q.H.P.; Software, Q.H.P. and T.D.P.; Visualization, V.D.P.; Writing—original draft, V.T.D.; Writing—review & editing, H.-N.N.

**Funding:** This research was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) gran<sup>t</sup> number 107.02-2018.30.

**Conflicts of Interest:** The authors declare no conflict of interest.
